Equation  shows importance re-weighting used on trajectories sampled from \(\pi _b\)  in order to obtain OPE estimates corresponding to our target population-level estimand. From the derivation, we re-iterate and highlight assumptions [1]}. First, Equation REF  implies a factorization of the data-generating process which corresponds to the MDP assumption in Figure REF . Equation , replaces interventional random variables with observed counterparts due to sequential ignorability, which follows from the structural assumption of the MDP and the backdoor-criterion [2]}: \(Y_{t}^{a_t}, Y_{t}^{a^{\prime }_t}, \cdots \perp A_t \vert \mathcal {H}_{t}\) . Finally, Equation , is estimable due to assumptions of overlap: that the support of \(\pi _e(A_t| S_t) > 0\)  wherever \(\pi _b(A_t| S_t)> 0\) . More specifically, when we assume ignorability conditioning to \(S_{t}\) , that is, \(Y_{t}^{a_t}, Y_{t}^{a^{\prime }_t}, \cdots \perp A_t \vert S_{t}\) , we assume that the underlying DGP corresponds to the MDP as in Figure REF  (left). When these assumptions hold, \(\mathbb {E}[G(\mathcal {H}^{\pi _e})|do(\pi _e), \mathcal {H}_{T}^{\pi _b}]\)  is estimable as we show above and IS provides an unbiased estimate of our OPE estimand.
